An Effective Meshless Approach for Inverse Cauchy Problems in 2D and 3D Electroelastic Piezoelectric Structures

In the past decade,notable progress has been achieved in the development of the generalized finite difference method(GFDM).The underlying principle of GFDM involves dividing the domain into multiple sub-domains.Within each sub-domain,explicit formulas for the necessary partial derivatives of the partial differential equations(PDEs)can be obtained through the application of Taylor series expansion and moving-least square approximation methods.Consequently,the method generates a sparse coefficient matrix,exhibiting a banded structure,making it highly advantageous for large-scale engineering computations.In this study,we present the application of the GFDM to numerically solve inverse Cauchy problems in two-and three-dimensional piezoelectric structures.Through our preliminary numerical experiments,we demonstrate that the proposed GFDMapproach shows great promise for accurately simulating coupled electroelastic equations in inverse problems,even with 3%errors added to the input data.

ELECTROELASTIC INTENSIFICATION NEAR ANTI-PLANE CRACK IN A FUNCTIONALLY GRADIENT PIEZOELECTRIC CERAMIC STRIP

Following the theory of linear piezoelectricity,we consider the electro-elastic prob- lems of a finite crack in a functionally gradient piezoelectric ceramic strip.By the use of Fourier transforms we reduce the problem to solving two pairs of dual integral equations.The solution to the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind.Numerical calculations are carried out for piezoelectric ceramics.The electric field intensity factors and the energy release rate are shown graphically,and the electroelastic interactions are illustrated.

THE EFFECTIVE PROPERTIES OF PIEZOCOMPOSITES, PART Ⅱ: THE EFFECTIVE ELECTROELASTIC MODULI

Since piezoelectric ceramic/polymer composites have been widely used as smart materials and smart structures, it is more and more important to obtain the closed-from solutions of the effective properties of piezocomposites with piezoelectric ellipsoidal inclusions. Based on the closed-from solutions of the electroe- lastic Eshelby's tensors obtained in the part I of this paper and the generalized Bu- diansky's energy-equivalence framework, the closed-form general relations of effective electroelastic moduli of the piezocomposites with piezoelectric ellipsoidal inclusions are given. The relations can be applicable for several micromechanics models, such as the dilute solution and the Mori-Tanaka's method. The difference among the various models is shown to be the way in which the average strain and the average electric field of the inclusion phase are evaluated. Comparison between predicted and exper- imental results shows that the theoretical values in this paper agree quite well with the experimental results. These expressions can be readily utilized in analysis and design of piezocomposites.

ELECTROELASTIC FIELD FOR AN IMPERMEABLE ANTI-PLANE SHEAR CRACK IN A PIEZOELECTRIC CERAMICS PLATE

Electroelastic behavior of a cracked piezoelectric ceramics plate subjected to four Cases of combined mechanical-electrical loads is analyzed. The integral transform method is applied to convert the problem involving an impermeable anti-plane crack to dual integral equations. Solving the resulting equations, the explicit analytic expressions for electroelastic field along the crack line and the intensity factors of relevant quantities near the crack tip and the mechanical strain energy release rate we obtained, The known results for an infinite piezoelectric ceramics plane containing an impermeable anti-plane crack are recovered from the present results only if the thickness of the plate h --> infinity.

TWO-DIMENSIONAL ELECTROELASTIC FUNDAMENTAL SOLUTIONS FOR GENERAL ANISOTROPIC PIEZOELECTRIC MEDIA

Explicit fomulas for 2-D electroelastic fundamental solutions in general anisotropic piezoelectric media subjected to a line force and a line charge are obtained by using the plane wave decomposition method and a subsequent application of the residue calculus. 'Anisotropic' means that any material symmetry restrictions are not assumed. 'Two dimensional' includes not only in-plane problems but also anti-plane problems and problems in which in-plane and anti-plane deformations couple each other. As a special case, the solutions for transversely isotropic piezoelectric media are given.

Some variational principles in electroelastic media under finite deformation

It is proposed that the universal thermodynamic energy variational principle is in-cluded in the first law of thermodynamics. Some variational principles in the elec-troelastic media under finite deformation are derived from this universal thermo-dynamic variational principle. It is suggested that in the general electroelastic analysis the environment should be considered together with the discussed elec-troelastic medium. For the variational principle of nonlinear electroelastic media the variation of the electric potential is coupled with the virtual displacement,and the variation of the initial volume should be considered. The Maxwell stress in the initial configuration is naturally derived from this variational principle and it is unique in the second order precision.

Electroelastic field induced by thin interface electrodes between two bonded dissimilar piezoelectric ceramics

The electroelastic analysis of multiple collinear electrodes embedded at the interface of two bonded dissimilar piezoelectric ceramics is made. Within the framework of linear piezoelectricity, the Fourier transform technique is applied to reducing the problem to a singular integral equation with Cauchy kernel. Two particular cases are especially emphasized. For a single interface electrode, the electroelastic field is obtained in the entire plane of a two-phase piezoelectric composite in terms of elementary functions. For two collinear interface electrodes of equal length, a closed-form solution is determined along the interface. Obtained results reveal that near the electrode edges, the induced electroelastic field exhibits a square-root singularity, and the oscillatory singularity does not occur for arbitrary two piezoelectric ceramics poled in the same or opposite directions normal to the interface. Across the electrode, the normal component of stress is continuous, while that of strain exhibits a jump, implying strain incompatibility due to the mismatch of the material properties of two dissimilar poled ceramics.

Electroelastic Analysis of Two-Dimensional Piezoelectric Structures by the Localized Method of Fundamental Solutions

Accurate and efficient analysis of the coupled electroelastic behavior of piezoelectric structures is a challenging task in the community of computational mechanics.During the past few decades,the method of fundamental solutions(MFS)has emerged as a popular and well-established meshless boundary collocation method for the numerical solution of many engineering applications.The classical MFS formulation,however,leads to dense and non-symmetric coefficient matrices which will be computationally expensive for large-scale engineering simulations.In this paper,a localized version of the MFS(LMFS)is devised for electroelastic analysis of twodimensional(2D)piezoelectric structures.In the LMFS,the entire computational domain is divided into a set of overlapping small sub-domains where the MFS-based approximation and the moving least square(MLS)technique are employed.Different to the classical MFS,the LMFS will produce banded and sparse coefficient matrices which makes the method very attractive for large-scale simulations.Preliminary numerical experiments illustrate that the present LMFM is very promising for coupled electroelastic analysis of piezoelectric materials.

Differential derivation of momentum and energy equations in electroelasticity

This paper presents a derivation of the equations of linear momentum, angular momentum, and energy of an electroelastic body using a composite particle consisting of two differential elements based on Tiersten＇s two-continuum model. The differential derivation shows the physics involved in a way different from the integral approach in the literature. Like the integral approach, it also produces the expressions of the electric body force, couple, and power which are fundamental to the development of the nonlinear macroscopic theory of an electroelastic body.

ON THE DERIVATION OF ELECTRIC BODY FORCE,COUPLE AND POWER IN AN ELECTROELASTIC BODY

This paper presents a procedure for the derivation of the expressions for electric body force, couple, and power in a nonlinear electroelastic body under electromechanical loads. The derivation is based on Tierseten＇s two-continuum model but much simplified.

GENERAL COUPLED SOLUTION OF ANISOTROPIC PIEZOELECTRIC MATERIALS WITH AN ELLIPTIC INCLUSION

In this investigation,the Stroh formalism is used to develop a general solution for an infinite,anisotropic piezoelectric medium with an elliptic inclusion. The coupled elastic and electric fields both inside the inclusion and on the interface of the inclusion and matrix are given.

ANISOTROPIC THERMOPIEZOELECTRIC SOLIDS WITH AN ELLIPTIC INCLUSION OR A HOLE UNDER UNIFORM HEAT FLOW

The two-dimensional problem of a thermopiezoelectric material containing an elliptic inclusion or a hole subjected to a remote uniform heat flow is studied. Based on the extended Lekhnitskii formulation for thermopiezoelectricity, conformal mapping and Laurent series expansion, the explicit and closed-form solutions are obtained both inside and outside the inclusion (or hole). For a hole problem, the exact electric boundary conditions on the hole surface are used. The results show that the electroelastic fields inside the inclusion or the electric field inside the hole are linear functions of the coordinates. When the elliptic hole degenerates into a slit crack, the electroelastic fields and the intensity factors are obtained. The effect of the heat how direction and the dielectric constant of air inside the crack on the thermal electroelastic fields are discussed. Comparison is made with two special cases of which the closed solutions exist and it is shown that our results are valid.

THE EFFECTIVE PROPERTIES OF PIEZOCOMPOSITES, PART Ⅰ: SINGLE INCLUSION PROBLEM

The problem of a piezoelectric ellipsoidal inclusion in an infinite non- piezoelectric matris is very important in engineering. In this paper, it is solved via Green's function technique. The closed-form solutions of the electroelastic Eshelby's tensors for this kind of problem are obtained. The electroelastic Eshelby's tensors can be expressed by the Eshelby's tensors of the perfectly elastic inclusion problem and the perfectly dielectric inclusion problem. Since the closed-form solutions of the Eshelby's tensors of the perfectly elastic inclusion problem and the perfectly dielectric inclusion problem can be given by theory of elasticity and electrodynamics, respectively, the electroelastic Eshelby's tensors can be obtained conveniently. Using these results, the closed-form solutions of the constraint elastic fields and the constraint electric fields inside the piezoelectric ellipsoidal inclusion are also obtained. These expressions can be readily utilized in solutions of numerous problems in the micromechanics of piezoelectric solids, such as the deformation and energy analysis, damage evolution and fracture of the piezoelectric materials.

Nonlinear dynamic analysis of dielectric elastomer membrane with electrostriction

The dielectric elastomer(DE)is an important intelligent soft material widely used in soft actuators,and the dynamic response of the DE is highly nonlinear due to the material properties.In the DE,electrostriction denotes the deformation-dependent permittivity.In the present study,we formulate the nonlinear dynamic governing equations of the DE membrane considering the electrostriction effect.The free vibration and parametric excitation of the DE membrane with different geometric sizes are calculated.The free vibration bifurcations induced by the initial location and the voltage are both discussed according to an energy-based approach.The amplitude-frequency characteristics and bifurcation diagrams of parametric excitation are also given.The results show that electrostriction decreases the free vibration amplitude and increases the frequency,but it has less influence on the parametric excitation oscillation frequency and decreases the parametric excitation amplitude except when the membrane resonates.The initial location and the applied voltage can induce the snap-through instability of the free vibration.A large geometric size will lead to a much lower resonance frequency.The resonance amplitudes increase while the resonance frequencies decrease with the increase in the applied voltage.The critical voltage of snap-through instability for the parametric excitation is larger than that for the free vibration one.

Bleustein-Gulyaev SAWS with Low Losses: Approximate Direct Solution

The main properties (attenuation along the surface, attenuation in depth, additional radiation in depth, dispersion in propagation space) of Bleustein-Gulyaev surface acoustic waves (SAWs) in electroelasticity are determined in terms of a perturbation due to viscosity. This paves the way for a study of the perturbed motion of associated quasi-particles in the presence of low losses.

WAVES IN PRE-STRETCHED INCOMPRESSIBLE SOFT ELECTROACTIVE CYLINDERS:EXACT SOLUTION

This paper studies wave propagation in a soft electroactive cylinder with an under- lying finite deformation in the presence of an electric biasing field. Based on a recently proposed nonlinear framework for electroelastieity and the associated linear incremental theory, the basic equations governing the axisymmetric wave motion in the cylinder, which is subjected to homo- geneous pre-stretches and pre-existing axial electric displacement, are presented when the elec- troactive material is isotropic and incompressible. Exact wave solution is then derived in terms of （modified） Bessel functions. For a prototype model of nonlinear electroactive material, illus- trative numerical results are given. It is shown that the effect of pre-stretch and electric biasing field could be significant on the wave propagation characteristics.

Bending analysis of a functionally graded piezoelectric cantilever beam

A new analysis based on Airy stress function method is presented for a functionally graded piezoelectric material cantilever beam. Assuming that the mechanical and electric properties of the material have the same variations along the thickness direction, a two-dimensional plane elasticity solution is obtained for the coupling electroelastic fields of the beam under different loadings. This solution will be useful in analyzing FGPM beam with arbitrary variations of material properties. The influences of the functionally graded material properties on the structural response of the beam subjected to different loads are also studied through numerical examples.